A bound for twists of \({\textrm{GL}}_3\times GL_2\) L-functions with composite modulus

Let \(\pi \) $\pi$ be a Hecke-Maass cusp form for \(\textrm{SL}_3({\textbf{Z}})\) ${\text{SL}}_3(\mathbf{Z})$ and let g be a holomorphic or Maass cusp form for \(\textrm{SL}_2({\textbf{Z}})\) ${\text{SL}}_2(\mathbf{Z})$ . Let \(\chi \) $\chi$ be a primitive Dirichlet character of modulus \(M=M_1M_2\) $M=M_1{M}_2$ with \(M_i\) $M_i$ prime, \(i=1,2\) $i=1,2$ . Suppose that \(M^{1/2+2\eta }<M_1<M^{1-2\eta }\) $M^{1/2+2\eta }<M_1<M^{1-2\eta }$ with \(0<\eta <1/8\) $0<\eta <1/8$ . Then we have \(\begin{aligned} L\left( \frac{1}{2},\pi \otimes g \otimes \chi \right) \ll _{\pi ,g,\varepsilon } M^{3/2-\eta +\varepsilon }. \end{aligned}\) $$\begin{array}{c}\hfill L\left(\frac{1}{2},,,\pi ,\otimes ,g,\otimes ,\chi \right){\ll }_{\pi ,g,\epsilon }{M}^{3/2-\eta +\epsilon }.\end{array}$$